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We consider NLS on $\T^2$ with multiplicative spatial white noise andnonlinearity between cubic and quartic. We prove global existence, uniquenessand convergence almost surely of solutions to a family of properly regularizedand renormalized approximating equations. In particular we extend a previousresult by A. Debussche and H. Weber available in the cubic and sub-cubicsetting.

Two dimensional nonlinear Schr\"odinger equation with spatial white noise potential and fourth order nonlinearity